Digraph from power mapping on noncommutative groups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050084
Author(s):  
Jinxing Zhao ◽  
Guixin Deng
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Cyclic Groups ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
Power Mapping

Let [Formula: see text] be a group and [Formula: see text] be a positive integer. The [Formula: see text]-power digraph [Formula: see text] is consisting of vertex set [Formula: see text] and there is a directed edge from [Formula: see text] to [Formula: see text] if and only if [Formula: see text]. We study the [Formula: see text]-power digraph on the semiproduct of cyclic groups. In particular, we obtain the distribution of indegree and cycles, and determine the structure of trees attached with vertices of power digraph. Finally, we establish a necessary and sufficient condition for isomorphism of digraphs [Formula: see text] and [Formula: see text].

scholarly journals Isomorphic Digraphs from Affine Maps of Finite Cyclic Groups

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Guixin Deng
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Cyclic Groups ◽  
Affine Maps ◽  
Necessary And Sufficient

Let be a positive integer. For any pair of integers and , let be the digraph whose set of vertices is , and there exists a directed edge from vertex to vertex if . In this paper, we obtain a necessary and sufficient condition for which .

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

On the Structure of G(H, k)

2014 ◽  
Vol 21 (02) ◽  
pp. 317-330 ◽  
Author(s):  
Guixin Deng ◽  
Pingzhi Yuan
Keyword(s):  
Abelian Group ◽  
Positive Integer ◽  
Explicit Formula ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Directed Edge ◽  
Necessary And Sufficient

Let H be an abelian group written additively and k be a positive integer. Let G(H, k) denote the digraph whose set of vertices is just H, and there exists a directed edge from a vertex a to a vertex b if b = ka. In this paper we give a necessary and sufficient condition for G(H, k1) ≃ G(H, k2). We also discuss the problem when G(H1, k) is isomorphic to G(H2, k) for a given k. Moreover, we give an explicit formula of G(H, k) when H is a p-group and gcd (p, k)=1.

scholarly journals Cyclic partitions of complete nonuniform hypergraphs and complete multipartite hypergraphs

2013 ◽  
Vol Vol. 15 no. 2 (Combinatorics) ◽  
Author(s):  
Shonda Gosselin ◽  
Andrzej Szymański ◽  
Adam Pawel Wojda
Keyword(s):  
Positive Integer ◽  
Nonempty Subset ◽  
Sufficient Conditions ◽  
Sufficient Condition ◽  
Uniform Hypergraph ◽  
Necessary And Sufficient Condition ◽  
Vertex Set ◽  
International Audience ◽  
Complete Multipartite ◽  
Necessary And Sufficient

Combinatorics International audience A \em cyclic q-partition of a hypergraph (V,E) is a partition of the edge set E of the form \F,F^θ,F^θ², \ldots, F^θ^q-1\ for some permutation θ of the vertex set V. Let Vₙ = \ 1,2,\ldots,n\. For a positive integer k, Vₙ\choose k denotes the set of all k-subsets of Vₙ. For a nonempty subset K of V_n-1, we let \mathcalKₙ^(K) denote the hypergraph ≤ft(Vₙ, \bigcup_k∈ K Vₙ\choose k\right). In this paper, we find a necessary and sufficient condition on n, q and k for the existence of a cyclic q-partition of \mathcalKₙ^(V_k). In particular, we prove that if p is prime then there is a cyclic p^α-partition of \mathcalK^(Vₖ)ₙ if and only if p^α + β divides n, where β = \lfloor \logₚ k\rfloor. As an application of this result, we obtain two sufficient conditions on n₁,n₂,\ldots,n_t, k, α and a prime p for the existence of a cyclic p^α-partition of the complete t-partite k-uniform hypergraph \mathcal K^(k)_n₁,n₂,\ldots,n_t.

A study on distance in graph complement

2020 ◽  
Vol 12 (03) ◽  
pp. 2050045
Author(s):  
A. Chellaram Malaravan ◽  
A. Wilson Baskar
Keyword(s):  
Positive Integer ◽  
Complete Graph ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient

The aim of this paper is to determine radius and diameter of graph complements. We provide a necessary and sufficient condition for the complement of a graph to be connected, and determine the components of graph complement. Finally, we completely characterize the class of graphs [Formula: see text] for which the subgraph induced by central (respectively peripheral) vertices of its complement in [Formula: see text] is isomorphic to a complete graph [Formula: see text], for some positive integer [Formula: see text].

Generalized unit and unitary Cayley graphs of finite rings

2019 ◽  
Vol 18 (01) ◽  
pp. 1950006 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Anukumar Kathirvel
Keyword(s):  
Commutative Ring ◽  
Undirected Graph ◽  
Unit Element ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Finite Rings ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient ◽  
Nonzero Identity

Let [Formula: see text] be a finite commutative ring with nonzero identity and [Formula: see text] be the set of all units of [Formula: see text] The graph [Formula: see text] is the simple undirected graph with vertex set [Formula: see text] in which two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if there exists a unit element [Formula: see text] in [Formula: see text] such that [Formula: see text] is a unit in [Formula: see text] In this paper, we obtain degree of all vertices in [Formula: see text] and in turn provide a necessary and sufficient condition for [Formula: see text] to be Eulerian. Also, we give a necessary and sufficient condition for the complement [Formula: see text] to be Eulerian, Hamiltonian and planar.

THE ITERATION DIGRAPHS OF GROUP RINGS OVER FINITE FIELDS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350162 ◽  
Author(s):  
YANGJIANG WEI ◽  
GAOHUA TANG ◽  
JIZHU NAN
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Directed Edge ◽  
Cycle Structure ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cn over a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCn and k such that G(𝔽prCn, k) is semiregular.

scholarly journals A generalization of the practical numbers

2018 ◽  
Vol 14 (05) ◽  
pp. 1487-1503
Author(s):  
Nicholas Schwab ◽  
Lola Thompson
Keyword(s):  
Positive Integer ◽  
Arithmetic Function ◽  
Arithmetic Functions ◽  
Asymptotic Density ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Necessary And Sufficient ◽  
First Time

A positive integer [Formula: see text] is practical if every [Formula: see text] can be written as a sum of distinct divisors of [Formula: see text]. One can generalize the concept of practical numbers by applying an arithmetic function [Formula: see text] to each of the divisors of [Formula: see text] and asking whether all integers in a certain interval can be expressed as sums of [Formula: see text]’s, where the [Formula: see text]’s are distinct divisors of [Formula: see text]. We will refer to such [Formula: see text] as “[Formula: see text]-practical”. In this paper, we introduce the [Formula: see text]-practical numbers for the first time. We give criteria for when all [Formula: see text]-practical numbers can be constructed via a simple necessary-and-sufficient condition, demonstrate that it is possible to construct [Formula: see text]-practical sets with any asymptotic density, and prove a series of results related to the distribution of [Formula: see text]-practical numbers for many well-known arithmetic functions [Formula: see text].

Pairs of Consecutive Power Residues or Non-Residues

1964 ◽  
Vol 16 ◽  
pp. 310-314 ◽  
Author(s):  
J. H. Jordan
Keyword(s):  
Positive Integer ◽  
Cyclic Group ◽  
Proper Subgroup ◽  
Multiplicative Group ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Root Of Unity ◽  
Necessary And Sufficient

For a positive integer k and a prime p ≡ 1 (mod k), there is a proper subgroup, R, of the multiplicative group (mod p) consisting of the kth power residues (mod p). A necessary and sufficient condition that an integer t be an element of R is that the congruence xk ≡ t (mod p) be solvable. The cosets, not R, formed with respect to R are called classes of kth power nonresidues, and form with R a cyclic group of order k. Let ρ be a primitive kth root of unity and let S be a class of non-residues that is a generator of this cyclic group. There is a kth power character X (mod p) such that

REGULARITY AND STRONG REGULARITY IN THE CONTEXT OF CERTAIN CLASSES OF RINGS

2013 ◽  
Vol 12 (05) ◽  
pp. 1250205 ◽  
Author(s):  
MICHAŁ ZIEMBOWSKI
Keyword(s):  
Positive Integer ◽  
Sufficient Condition ◽  
Strong Regularity ◽  
Necessary And Sufficient Condition ◽  
Weak Dimension ◽  
Ring Of Polynomials ◽  
Necessary And Sufficient ◽  
Semihereditary Rings ◽  
The Ideal

We consider the ring R[x]/(xn+1), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xn+1) is the ideal of R[x] generated by xn+1 and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xn+1) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

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